W' balance as a determinant of pacing in individual pursuit: application of Skiba's dynamic model to anaerobic capacity management

The interindividual variability of performance in 4 km individual pursuit is not primarily explained by differences in sustainable critical power but by the dynamic management of the anaerobic energy reservoir termed W' (Skiba et al., 2012). An athlete registering 4:12 in their first attempt and experiencing a chronometric collapse to 4:22 two weeks later has not substantially modified their physiological capacity: they have altered the temporal profile of W' depletion. Formalising this depletion constitutes the scientific basis of modern pacing in events shorter than five minutes.

Theoretical framework: the Monod and Scherrer hyperbolic model

The two-parameter model proposed by Monod and Scherrer (1965) establishes a hyperbolic relationship between sustainable power and time to exhaustion in high-intensity exercise:

In this formulation, CP (critical power) represents the horizontal asymptote of the power-time curve and corresponds to the power theoretically sustainable in steady state at the expense exclusively of aerobic metabolism. The parameter W' (W prime) represents the finite energy available above CP, expressed in joules, and constitutes the anaerobic reservoir mobilisable during supracritical exercise.

Skiba et al. (2012) transformed this static formulation into a dynamic description of the instantaneous W' reservoir balance. When applied power exceeds CP, W' depletes at a rate of (P − CP) joules per second; when power drops below CP, W' recharges with a time constant dependent on the instantaneous recovery level. Depletion kinetics are fast; recharge kinetics, slow.

Mathematical formulation of dynamic W' balance

The first equation integrates the net depletion of the W' reservoir throughout exercise, weighted by an exponential decay function f. The second equation specifies the time constant τ, dependent on the difference between CP and mean power during subcritical intervals. This constant determines the W' replenishment speed during relative recovery phases.

Application to the individual pursuit regime

During a 4 km IP event, applied power is systematically maintained in the 108-115% CP range across the four complete minutes. Consequently, W' balance evolves monotonically decreasing without recharge phases: the only open question is the depletion rate.

An international-level cyclist exhibits characteristic values of CP ≈ 400 W and W' ≈ 22,500 J. If a constant power of 445 W is sustained, cumulative expenditure amounts to 45 W × 240 s = 10,800 J, leaving a reserve of 11,700 J: the race has been executed below potential. If mean power rises to 465 W, expenditure reaches 15,600 J and final reserve reduces to 6,900 J. At a constant 500 W, theoretical expenditure (24,000 J) exceeds available W': the anaerobic system exhausts before lap 15 is completed, with phenotypic manifestation as sudden velocity loss from second 195 onwards.

Cost of the start phase on the W' budget

The first 15 seconds of an individual pursuit constitute a pure sprint with instantaneous mean power above 700 W. This effort demands between 4,500 and 6,000 J from the W' reservoir depending on athlete profile. Systematic omission of this cost in pacing planning leads to underestimation of real depletion and manifestation of collapse at kilometre three. The operational axiom is stated as follows: the start does not constitute a decision variable but a fixed cost of the event; the budget available for the cruise phase is defined as total W' minus start cost.

Quantitative example of W' expenditure distribution

An athlete with CP = 400 W and W' = 22,500 J is considered in the IP 4 km event:

This profile terminates with a reserve of 6,000 J, indicating that exercise has been executed with margin relative to theoretical potential. The optimal reserve interval at the finish line falls between 1,500 and 3,000 J. A null reserve predicts collapse in the final lap with high probability; a reserve above 5,000 J indicates conservative pacing and avoidable chronometric loss.

Field estimation protocol for CP and W'

The standard protocol in the absence of laboratory instrumentation consists of a 3-minute maximum test following a structured 20-30 minute warm-up. Mean power registered during the last 30 seconds approximates CP with characteristic error of ±5%. To increase precision, the double protocol is recommended:

Alternatively, the combination of a 3-minute test and a 12-minute test separated by 48 hours allows least-squares fitting of the power-time hyperbola. Characteristic fitting precision reaches ±2% for CP and ±8% for W', values compatible with operational pacing programming.

References: Skiba, P. F., Chidnok, W., Vanhatalo, A., & Jones, A. M. (2012). Modeling the expenditure and reconstitution of work capacity above critical power. Medicine and Science in Sports and Exercise, 44(8), 1526-1532. Monod, H., & Scherrer, J. (1965). The work capacity of a synergic muscular group. Ergonomics, 8(3), 329-338. Jones, A. M., Burnley, M., Black, M. I., et al. (2019). The maximal metabolic steady state. Frontiers in Physiology, 10, 33. Corbett, J. (2009). An analysis of the pacing strategies adopted by elite athletes during track cycling. International Journal of Sports Physiology and Performance, 4(2), 195-205. Coakley, S. L., & Passfield, L. (2018). Cycling performance is superior for time-to-exhaustion versus time-trial. European Journal of Applied Physiology, 118, 1541-1549.